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Mathematics-Online course: Linear Algebra - Basic Structures - Groups and Fields

Galois Fields

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The field GF[$ 2^2$] with $ 4$ elements can be given by the operation tables of addition and multiplication:

$ +$ 0 $ 1$ $ a$ $ b$
0 0 $ 1$ $ a$ $ b$
$ 1$ $ 1$ 0 $ b$ $ a$
$ a$ $ a$ $ b$ 0 $ 1$
$ b$ $ b$ $ a$ $ 1$ 0
$ \cdot$ 0 $ 1$ $ a$ $ b$
0 0 0 0 0
$ 1$ 0 $ 1$ $ a$ $ b$
$ a$ 0 $ a$ $ b$ $ 1$
$ b$ 0 $ b$ $ 1$ $ a$

The underlying construction principle found by Galois is not as simple as that of prime fields $ \mathbb{Z}_p$. It applies to the construction of fields with $ p^\ell$, $ \ell\in\mathbb{N}$ elements for any prime number $ p$. It can be proved that all finite fields can be obtained that way.

  automatically generated 4/21/2005